Derivada de una función polinomial (paso a paso) 👌👆. Cálculo diferencial. 2

Summary

This video explains how to differentiate polynomial functions using the power rule. It demonstrates the process step-by-step, starting with a cubic term and moving through higher-degree terms. The instructor shows how to apply the rule to terms like x³, 5x²⁰, and 2x, emphasizing how the exponent decreases and the coefficient multiplies. Additionally, the video covers the derivative of fractional terms like x/5 and polynomial terms with lower exponents, offering clear examples and reinforcing key concepts. The approach provides a practical and easy-to-understand method for handling derivatives of polynomials.

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Q & A

What is the general process of differentiating polynomial functions?

The process of differentiating polynomial functions involves applying the power rule. For each term in the polynomial, you multiply the coefficient by the exponent, then reduce the exponent by 1.

How do you differentiate a term like x^3?

To differentiate a term like x^3, you drop the exponent (3), multiply it by the coefficient (which is 1 for x^3), and then reduce the exponent by 1. The result is 3x^2.

What happens when differentiating a term like 2x?

When differentiating a term like 2x, the exponent of x is 1. According to the power rule, the derivative of x is just the coefficient, so the result is 2.

What is the derivative of 5x^20?

The derivative of 5x^20 involves multiplying the coefficient (5) by the exponent (20), which gives 100, and then reducing the exponent by 1. The result is 100x^19.

How do you differentiate a fraction like x/5?

To differentiate x/5, treat it as x raised to the power of 1, multiplied by 1/5. Since the exponent is 1, the derivative is simply the coefficient (1/5).

What does the power rule state for differentiating polynomials?

The power rule states that to differentiate a term of the form x^n, multiply the exponent (n) by the coefficient, then reduce the exponent by 1. The formula is: d/dx(x^n) = n * x^(n-1).

How do you apply the power rule to a term like 7x^4?

For the term 7x^4, you multiply the coefficient (7) by the exponent (4), giving 28, and then reduce the exponent by 1, resulting in 28x^3.

When differentiating a term like 2x, why does the x disappear?

When differentiating 2x, the exponent of x is 1. According to the power rule, the derivative of x is just 1, so only the coefficient remains, which is 2 in this case.

What is the derivative of the function f(x) = x^3 + 5x^20 + 2x?

The derivative of f(x) = x^3 + 5x^20 + 2x is f'(x) = 3x^2 + 100x^19 + 2.

How do you differentiate the term 7x^4 in the expression f(x) = 7x^4?

To differentiate 7x^4, apply the power rule: multiply the exponent (4) by the coefficient (7) to get 28, then reduce the exponent by 1. The result is 28x^3.